{
  "abstract": "We determine which homotopy nine-spheres admit positive sectional curvature: they are precisely the four smooth types with vanishing real Dirac index. The other four admit no metric of nonnegative sectional curvature. In particular, we construct a strictly positively curved metric on the exotic Kervaire nine-sphere, answering the longstanding nonnegative-curvature existence question in this dimension. The proof keeps explicit attaching maps throughout. A quaternionic commutator realizes the Kervaire sphere, a mixed representation realizes the Hopf image of the exotic eight-sphere, and a four-periodic normal-framing formula identifies the remaining index-zero class. Following the compatible-disk method, we give the curvature and boundary estimates for the first two models, including an explicit lower bound for the Kervaire metric in terms of the curvature and first covariant-derivative norms of a fixed connection, with the base scale and fiber normalization fixed. For the third model, we construct a positive metric on a cylinder joining a quotient metric to its pullback by a marked right Hopf twist. A four-form modification controls the connection terms involving both left and right quaternionic multiplication. Two positive disk fillings close the cylinder with the prescribed attaching map. A local convolution lemma supplies the smooth gluing while preserving a strict sectional-curvature lower bound.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.03298",
  "author_total": 4,
  "authors": [
    {
      "name": "Zhiqi Chen"
    },
    {
      "name": "Shaoqiang Deng"
    },
    {
      "name": "Junna Ni"
    },
    {
      "name": "Hui Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/d1fvbvyncg04m.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.03298",
  "fulltext_url": null,
  "id": "d1fvbvyncg04m",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.03298",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.DG"
    ],
    [
      "openalex",
      "3103|Advanced Differential Geometry Research"
    ]
  ],
  "openalex": "W7221789106",
  "pdf_url": "https://arxiv.org/pdf/2610.03298",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/d1fvbvyncg04m.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.03298",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.03298"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Positive sectional curvature on homotopy nine-spheres",
  "title_text": "Positive sectional curvature on homotopy nine-spheres",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/d1fvbvyncg04m",
  "venue": "arXiv",
  "versions": []
}